Toward a Systematic Approach to the Economic Effects of Risk: Characterizing Utility Functions

• Date: 01 June 2018
• Author: Christian Gollier,Miles S. Kimball
• Published date: 01 June 2018
• DOI: http://doi.org/10.1111/jori.12249

©2018 The Journal of Risk and Insurance. Vol.85, No. 2, 397–430 (2018).

DOI: 10.1111/jori.12249

Toward a Systematic Approach to the Economic

Effects of Risk: Characterizing Utility Functions

Christian Gollier

Miles S. Kimball

Abstract

The diffidence theorem, together with complementary tools, can aid in illu-

minating a broad set of questions about how to mathematically characterize

the set of utility functions with specified economic properties. This article es-

tablishes the technique and illustrates its application to many questions, old

and new. For example, among many other older and other technically more

difficult results, it is shown that (1) several implications of globally greater

risk aversion depend on distinct mathematical properties when the initial

wealth level is known, (2) whether opening up a new asset market increases

or decreases saving depends on whether the reciprocal of marginal utility is

concave or convex, and (3) whether opening up a new asset market raises

or lowers risk aversion toward small independent risks depends on whether

absolute risk aversion is convex or concave.

Introduction

By implying linearity of preferences in probabilities, von Neumann–Morgenstern ex-

pected utility theory provides a great deal of common structure for applied problems.

Yetthe literature on the economics of risk has often approached each applied problem

or small class of applied problems in what looks like very different ways. In this ar-

ticle, we endeavor to make a bare beginning to systematizing approaches to applied

problems in the economics of risk. In particular, we study a systematic technique

for approaching the mathematical characterization of sets of utility functions with

particular economic properties. That is, the objective here is to build a path between

economic properties of preferences and mathematical properties of utility functions.

In a companion paper, “New Methods in the Classical Economics of Uncertainty:

Comparing Risks” (Gollier and Kimball, 2018), we address the mathematical charac-

terization of risks that have particular economic properties in relation to specified sets

of utility functions.

A spur for the development of more systematic techniques in the economics of risk is

the burgeoning and technically difficult literature on backgroundrisks. As Kihlstrom,

Christian Gollier is at the Toulouse School of Economics. Gollier can be contacted via e-mail:

christian.gollier@tse-fr.eu.Miles S. Kimball is at the University of Colorado Boulder. We would

like to acknowledge funding from the chair SCOR and FDIR at TSE.

397

398 The Journal of Risk and Insurance

Romer, and Williams (1981), Ross (1981), Doherty and Schlesinger (1983), and Pratt

and Zeckhauser (1987), our understanding of the behavior of agents facing more than

one source of risk has made a great deal of progress. A common feature of analyses

of the interaction between independent risks is the existence of paradoxes that can be

solved by putting additional restrictions on the utility function. For example, Kimball

(1993), Eeckhoudt, Gollier, and Schlesinger (1996), and Gollier and Pratt (1996) show

conditions under which adding a nonmarketable zero-mean risk to wealth can induce

a risk-averse agent to purchase more of another independent risky asset. Eeckhoudt

and Schlesinger (2006) push this literature further.

Difficult background risk problems and more basic problems in the economics of risk

often share elements with the same technical structure. More than 20 years ago, we

wrote an early version of this article (Gollier and Kimball, 1994) in which we provided

a simple method for solving these problems. The centerpiece of this method is the

“diffidence theorem.” This theorem has since been used by various authors to solve

some of these complex problems in a simple way, as we show in examples throughout

this article.

In addition to illustrating a technique of wide applicability, reframing many existing

results in useful ways, and establishing many minor new results, we establish at least

three important substantive results not in the previous literature. First, we clarify the

ways in which globally greater risk aversion can be overkill for an applied problem. If

something is known about initial starting wealth, the much weaker condition of cen-

trally greater risk aversion or the even weaker condition of centrally greater diffidence

will often be enough. This leads to a much deeper understanding of the Arrow–Pratt

theory. The mathematics is so basic, it is often tucked, in some form, somewhere into

proofs in previous papers, but has not been explored fully in its own right. Second,

we give a mathematical characterization of the set of utility functions for which open-

ing up a new asset market would lead to less saving, and the antiphonal set of utility

functions for which opening up a new asset market would lead to more saving. Third,

we characterize the set of utility functions for which opening up a new asset market

would make an agent more risk averse toward a small risk independent of the risk in

the new asset market.

Let us turn now to the diffidence theorem itself. In many instances, the compara-

tive statics problem involving risk can be written in the following form: under what

conditions on functions f1and f2can we guarantee that

∀˜

x:Ef

1(˜

x)≤Ef

1(x0)=⇒ Ef

2(˜

x)≤Ef

2(x0), (1)

for some starting wealth w. Let us consider a very simple illustration of problem (1).

There are several ways to define the concept of “greater risk aversion.” For example,

one can stipulate the definition that an agent with utility function u2is more risk

averse than another with utility function u1if and only if any risk ˜

xthat is rejected

by the latter is also rejected by the former, independent of the sure common wealth

level wof the two agents. This is an application of problem (1) with fi(x)=ui(w+x)

For each initial wealth level, this problem is equivalent to the following property:

∀˜

x:Eu1(w+˜

x)≤u1(x0)=⇒ Eu2(w+˜

x)≤u2(x0).(2)

Toward a Systematic Approach 399

The concepts of risk aversion, prudence, temperance, decreasing absolute risk aver-

sion, decreasing absolute prudence, properrisk aversion (Pratt and Zeckhauser, 1987),

risk vulnerability (Gollier and Pratt, 1996; Gollier, 2001), and standard risk aversion

(Kimball, 1993) can also be defined by using definition (1) for some specific pair

of functions (f1,f2), or by using its bivariate extension that we also present in this

article. In addition, we consider new concepts having the same structure as in (1).

For example, under what condition does an increase in nonmarketable background

risk raise the equilibrium risk free rate in the economy? Or, under what condition

does opening up a new asset market raise current consumption? And does open-

ing a market for a new asset reduce the demand for another independent risky as-

set? All these questions, and others, are solved in this article by using the diffidence

theorem.

The diffidence theoremis a consequence of the linearity of expected utility with respect

to probabilities. This linearity implies that condition (1) holds for all random variables

if and only if it holds for all “binary” random variables. This property singularly

simplifies the structure of the problem, and the diffidence theorem can be seen as an

application of that result.

Property (1) is related to the condition that f2is more concave that f1in the sense

of Arrow–Pratt. This is an obvious consequence of Jensen’s inequality, a ubiquitous

tool in decision theory under uncertainty. However, the “more concave” condition

is often too restrictive, and the diffidence theorem provides the right necessary and

sufficient condition for property (1). For example, when considering problem (2) for

a given wealth level w, it is easy to show that there exist pairs of functions (u1,u2)

for which u2is not more risk averse than u1in the sense of Arrow–Pratt, but still u2

rejects all lotteries that u1rejects. By using the diffidence theorem, we easily charac-

terize the notion of “centrally greater diffidence,” which is the weakest condition on

(u1,u2) that guarantees (2), and lends the diffidence theorem its name because it is

the most straightforward application of the theorem. Of course, because this property

must hold for small risks, a necessary condition for centrally greater diffidence is that

−u

2(x0)/u

2(x0) be larger than −u(x0)/u

1(x0), but this condition need not hold at other

wealth levels.

One of the beauties of the celebrated paper by Pratt (1964) is to show that a condition

“in the small,” that is, for small risks—if required to hold at all wealth levels, is

sufficient “in the large,” that is, for any risk. For example, if individual 2 dislikes all

small risks that individual 1 dislikes, and if this is true at all common wealth levels,

then it is known that individual 2 also dislikes any risk that individual 1 dislikes. Such

a result is not true in general under the structure (1), except in some special cases that

we identify in this article.

In the “The Diffidence Theorem” section, we prove the diffidence theorem and its

corollary. We illustrate the use of the diffidence theorem by presenting many appli-

cations in the “Applications of the Diffidence Theorem” section. The “The Bivariate

Diffidence Theorem” section is devoted to the bivariate diffidence theorem and its

applications. Other extensions are considered in the “Other Extensions” section. The

“Concluding Remarks” section concludes.